The free summer signing from Everton, whose thumping first-half header hit the crossbar, chested home Leeds' third goal soon after the restart after quickfire goals from Jayden Bogle and Noah Okafor had put them in control at half-time.
Forest substitute Lorenzo Lucca marked his debut for the club by heading home a late consolation as they slipped to their 13th league defeat of the season.
18th-placed West Ham won at fellow strugglers Burnley on Saturday to close the gap back to six points with Leeds.
But that result is more worrying for Forest, who are now just three points above the danger zone.
Leeds never looked back after taking a 26th-minute lead on Friday night, roared on by their fans to a fifth win of the season under the Elland Road floodlights and their sixth at home in total.
Daniel Farke's side went straight on the offensive from kick-off, with Forest defender Zach Abbott, making his full top-flight debut, slicing Gabriel Gudmundsson's low cross narrowly over the crossbar.
The visitors carried a threat on the break though and after comfortably holding Callum Hudson-Odoi's low shot, Leeds goalkeeper Karl Darlow produced a superb save to keep out Nicolas Dominguez's effort.
Igor Jesus headed just wide soon after as Forest grew into the game before Calvert-Lewin's towering header crashed down off the underside of the crossbar.
Leeds then seized full control with two goals in four minutes just before the half-hour mark.
Bogle stayed onside when running on to Ilia Gruev's beautifully-flighted through-ball and slid a composed low finish beneath Forest debutant Stefan Ortega for his first Premier League goal for Leeds.
The home side quickly doubled their advantage. Ortega kept out James Justin's shot and from the rebound the Leeds defender picked out Okafor, who buried his third goal of the season.
Noah Okafor roars with delight after scoring against Forest, Leeds going on to effectively put the game to bed by the 50-minute mark. (Image: Danny Lawson/PA Wire)
Forest delivered a warning when Darlow made another fine save to keep out Morgan Gibbs-White's fierce drive before the interval.
But there was no let-up from Leeds after the restart and they struck their third goal in the 49th minute.
Gruev was afforded far too much time on the right edge of the area and his curling cross to the near post was chested home by Calvert-Lewin to take the summer signing's league goal tally this season into double figures.
Leeds winger Daniel James made his first appearance since November off the bench after recovering from a hamstring injury.
Forest battled on and Morato's glancing header flew narrowly off target before substitute Lucca, a January signing from Napoli, headed home Omari Hutchinson's cross in the 86th minute.
Darlow made another fine save to turn Ibrahim Sangare's effort over the crossbar, but it was too little too late for the visitors.